Yes just probe around and see if theres any digital activity, nothing specific, your 'scope will suffice just as well as a 'probe, I find a probe being smaller easier to deal with and if you have/get one with tones you dont need to take your eyes off the pin being checked and risk shorting something.
I have some experience of aircrfaft instruments, if something is wrong or theres no comms the display usually shows something, though this is marine not aviation.
Yes thats a good idea, check to see if the lcd is driven by hardware, if so the pins to it will go to a driver ic, if not maybe the processor has a built in lcd driver, its unlikely the lcd will be totally driven by software, you should at least get the backplane squarewave, if you get plenty of squarewaves and some are out of phase with others then the unit is trying to display something.
Older lcd's 'wear' with sunlight, they fade, it unlikey this has faded to the point where you cant see it, has the unit been messed with?, there is a polarizing plate on the lcd that if removed or put back wrong you wont see anything, look at the lcd through a pair of polarized sunglasses (not a joke, when I was at school I modded a cheap digi watch so you could only see the digits through sunglasses).
Cant remember where or what but I sort of recall some lcd manufacturer having probelms with the adhesive used to bond the flexible pcb to the display, it reacted with the tracks on the glass and erroded them away, if this is the case then looks like its dead, you might be able to see it with a mini microscope.